Transverse Heteroclinic Bifurcation Of Piecewise Smooth Systems And Existence And Uniqueness Of Monotone Positive Solutions For Fractional Differential Equations

Many practical problems are affected not only by nonlinear factors,but also by nonsmooth factors due to collision,friction and switching.The corresponding mathematical models are piecewise smooth dynamical systems.Due to the nonsmoothness,even the simplest form of those systems,namely piecewise linear systems can exhibit very complicated dynamic behaviors.In addition to the typical bifurcations also occurring in smooth systems,the nonlinearities due to the discontinuities will also cause many new types of bifurcation phenomena,such as grazing,sliding and chattering that often lead to chaotic motions.Consequently,from both a theoretical and an application point of view,it is very important to study the chaotic behaviors in piecewise smooth systems.Previous studies shown that homoclinic and heteroclinic bifurcations are also important routes to chaos for piecewise smooth systems.It is well known that the classical Melnikov method is a powerful analytical tool for studying homoclinic and heteroclinic bifurcation of smooth systems.In recent years,many efforts have been made to extend the Melnikov method so it is also applicable to piecewise smooth systems,the high and infinite dimensional cases.In this paper,we study heteroclinic bifurcation and appearance of chaos in timeperturbed piecewise smooth hybrid systems with discontinuities on finitely many switching manifolds.We assume that the unperturbed system has a heteroclinic orbit connecting hyperbolic saddles of the unperturbed system that crosses every switching manifold transversally,possibly multiple times.We also assume that there is a reset map on each switching manifold.By using the method of Lyapunov-Schmidt,we obtain a set of Melnikov type functions.Then we prove that under certain conditions,the existence of simple zeros of those Melnikov type functions implies the occurrence of chaos of the system.Clearly,the system we considered is more general and the results obtained here can be applied to more general situations than those of in the work of Li and Du.As an application,we present an example of quasiperiodically excited system formed by two linked slender rocking blocks.Like integer differential equations,fractional differential equations are widely used in control,biology,signal and image processing,economics and other fields.The boundary value problem of fractional differential equations is an important problem in differential equation and has been extensively studied in recent years.Nevertheless,there are still many problems to be solved.In this paper,we discuss the existence and uniqueness of monotone positive solutions of the infinite-point boundary value problem of a class of higher-order nonlinear fractional differential equations.By analyzing the operator equation Tω+Sω=ω under given conditions,we obtain the existence and uniqueness of solutions of the problem by using the properties of Green’s function and the fixed point theorem.Our presentation is organized as follows.In Chapter 1,we summarize the progress on the study of bifurcation of piecewise smooth systems and boundary value problems of fractional differential equations and our main works.In Chapter 2,we present the preliminaries of piecewise smooth dynamics systems and fractional differential equations.In Chapter 3,we study heteroclinic bifurcation and chaos in time-perturbed piecewise smooth hybrid systems with discontinuities on finitely many switching manifolds.In Chapter 4 we will present an example of quasiperiodically excited system formed by two linked slender rocking blocks.In Chapter 5,we discuss the existence and uniqueness of monotone positive solutions of the infinite-point boundary value problem of a class of higherorder nonlinear fractional differential equations.Some concluding remarks are given in Chapter 6.